A function from a set AA to a set BB is then given by an operation, which is compatible with the equality (i.e. two elements which are equal in A are mapped to two elements which are equal in B), and is described as “a finite routine ff which assigns an element f(a)f(a) of BB to each given element aa of AA”. This notion of routine is left informal but must “afford an explicit, finite, mechanical reduction of the procedure for constructing f(a)f(a) to the procedure for constructing aa.”